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We study the homogeneous turbulence in the presence of a constant averagevelocity gradient in an infinite fluid domain, with a novel finite-scaleLyapunov analysis, presented in a previous work dealing with the homogeneousisotropic turbulence.  Here, the energy spectrum is studied introducing the spherical averaged paircorrelation function, whereas the anisotropy caused by the velocity gradient isanalyzed using the equation of the two points velocity distribution functionwhich is determined through the Liouville theorem. As a result, we obtain theevolution equation of this velocity correlation function which is shown to bevalid also when the fluid motion is referred with respect to a rotatingreference frame. This equation tends to the classical von K\'arm\'an-Howarthequation when the average velocity gradient vanishes.  We show that, the steady energy spectrum, instead of following the Kolmogorovlaw $\kappa^{-5/3}$, varies as $\kappa^{-2}$. Accordingly, the structurefunction of the longitudinal velocity difference $<\Delta u_r^n> \approxr^{\zeta_n}$ exhibits the anomalous scaling $\zeta_n \approx n/2$, and theintegral scales of the correlation function are much smaller than those of theisotropic turbulence.

Steady Homogeneous Turbulence in the Presence of an Average Velocity Gradient